Variance of Linear Combination in BSTS

Question

I'm studying Bayesian Structural Time Series package from Steven Scott (You can find it here BSTS).

I was looking at a simple derivation for an AR(1) model:

My question is simple: is there a mistake in the notes when specifying $Var(Y_t)$?

Answer 1

It seems like the slides you refer to includes a mistake. We can check this by calculating the variance of the $\operatorname{AR}(1)$ model. So let $Y_t = \phi Y_{t-1} + \varepsilon_t$ where $\lvert \phi\rvert < 1$ and $\{\varepsilon_t\}$ is an iid white noise process. Then

$$ \renewcommand{\var}{\operatorname{Var}} \var(Y_t) = \var(\phi Y_{t-1} + \varepsilon_t) = \phi^2\var(Y_{t-1}) + \var(\varepsilon_t). $$

As an $\operatorname{AR}(1)$ process with $\lvert\phi\rvert< 1$ is stationary, we have $\var (Y_t) = \var(Y_{t-1})$. Then $$ \var(Y_t) = \phi^2\var(Y_t) + \var(\varepsilon_t) \quad \iff \quad \var(Y_t) = \frac{\var(\varepsilon_t)}{1-\phi^2} $$ by rearranging.